# Lab 5 - Electron Charge-to-Mass Ratio

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1 Lab 5 Electron Charge-to-Mass Ratio L5-1 Name Date Partners Lab 5 - Electron Charge-to-Mass Ratio OBJECTIVES To understand how electric and magnetic fields impact an electron beam To experimentally determine the electron charge-to-mass ratio OVERVIEW In this experiment, you will measure e/m, the ratio of the electron s charge e to its mass m. Given that it is also possible to perform a measurement of e alone (the Millikan Oil Drop Experiment), it is possible to obtain the value of the mass of the electron, a very small quantity. If a particle carrying an electric charge q moves with a velocity v in a magnetic field B that is at a right angle to the direction of motion, it will experience the Lorentz force: F = qv B (5.1) Which, because of the vector product, is always perpendicular to both the magnetic field and the direction of motion. A constant force that is always perpendicular to the direction of motion will cause a particle to move in a circle. We will use this fact to determine e/m of the electron by measuring the radius of that circle. To this end we must: produce a narrow beam of electrons of known energy, produce a uniform magnetic field, find a way to measure the radius R of the circular orbit of the electrons in that magnetic field, and find the relation between that radius and the ratio e/m. We will discuss these tasks in order. The Electron Beam When one heats a piece of metal, say a wire, to 1,000 K or beyond, electrons will boil off from its surface. If one surrounds the wire with a positively charged electrode, an anode, the

2 L5-2 Lab 5 Electron Charge-to-Mass Ratio electrons will be attracted to it and move radially outward as indicated in Figure 5.1. On their way to the anode they will acquire a kinetic energy E k = 1 2 mv2 = ev, (5.2) where V is the potential difference, or voltage, between the heated filament, called the cathode, and the anode. Figure 5.1: Most of the electrons will strike the anode. However, if one cuts a narrow slit into the anode, those electrons that started out toward the slit will exit through it as a narrow beam with a kinetic energy E k. The Magnetic Field According to Ampere s law a wire carrying a current I is surrounded by a magnetic field B, as shown in Figure 5.2. If the wire is bent into a circle the field lines from all sides reinforce each other at the center, creating an axial field (see Figure 5.3). Usually one will not use a single loop of wire to create a field but a coil with many turns. ). Figure 5.2: Magnetic field of a straight wire Figure 5.3: Magnetic field of a wire loop

3 Lab 5 Electron Charge-to-Mass Ratio L5-3 If one uses two coaxial coils of radius R that are a distance d apart, as shown in Figure 5.4, the field at the center point between the coils will be nearly homogeneous 1. H. von Helmholtz ( ) realized that there remains a free parameter, namely the coil separation d, that can still be adjusted. He showed that when d = R, the result is a particularly homogeneous field in the central region between the coils. 2 Since that time Helmholtz coils have been used when there is a need for homogeneous magnetic fields. Figure 5.4: Magnetic field B of a pair of Helmholtz coils One can show that the field in the center of a Helmholtz coil is given by ( ) 8Nµ0 B = 5R I, (5.3) 5 where I is the current flowing through both coils, R is their mean radius, N is the number of turns of wire in each coil, and µ 0 = 4π 10 7 T m/a is the permeability constant. The Electron Orbit Experiments like the one that you will perform have been used to measure the mass of charged particles with great precision. In these experiments the particles move in a circular arc whose beginning and end are measured very accurately in a near perfect vacuum. For our simple experiment we cannot go to such lengths and a simple expedient has been used to make the electron orbit visible. The bulb surrounding the electron source is filled with helium vapor. When the electrons collide with the atoms, the atoms emit light so that one can follow the path of the electron beam. These collisions will diminish the accuracy of the experiment but it remains adequate for our purposes. A particle moving in a circle of radius R must be held there by a centripetal force F c = mv2 r. (5.4) In our case, that centripetal force is the Lorentz force, Equation (5.1), hence evb = mv2 (5.5) r This equation contains the velocity V, which we can eliminate by using Equation (5.2). Rewriting Equation (5.2), we find 1 The symmetry of the arrangement makes the first derivative of the field with respect to the axial direction vanish. 2 The second derivative vanishes as well.

4 L5-4 Lab 5 Electron Charge-to-Mass Ratio v = 2eV m (5.6) and hence e m = 2V B 2 r 2 = R 2 V (NIµ 0 r) 2 (5.7) In this equation, V is the voltage between cathode and anode and r is the mean radius of the circular electron orbit, both of which can be measured, and B is the magnetic field through which the electrons pass. We know the magnetic field at the center of the Helmholtz coil, which can be obtained, using Equation (5.3), from a measurement of the current through the coils, the dimension of the coils and the number of turns. The magnetic field does not change very much away from the center of the coils. INVESTIGATION 1: Finding e/m For this investigation, you will need the following: Helmholtz coil with e/m glass tube Bar magnet 3-m measuring tape Flashlight Incandescent light for reading manual Activity 1-1: The e/m Apparatus In this activity, you will familiarize yourself with the setup you will be using.

5 Lab 5 Electron Charge-to-Mass Ratio L5-5 Figure 5.5: e/m Apparatus Question 1-1: When you first observe the electron beam in this lab, the electrons will proceed straight down. As you turn up the current in the Helmholtz coils, the electrons will curve to the left. As you look at the apparatus shown in Figure 5, in what direction is the magnetic field, and in what describe the flow of current (positive) in the coils. 1. Turn the main power on. The unit will perform a self test lasting no more than 30 s. Do not do anything with the unit during the self test. When it is finished, the coil current display will be stabilized and indicate 000. The unit is now ready to use, but note that there is a ten minute warm-up time before you should take final measurements. You can go ahead and proceed with the remainder of this procedure. 2. Look in the center of the Helmholtz coils for the glass tube. The tube is extremely fragile, and the wires sticking out from the tube have current and voltage, so be very careful. This evacuated glass tube has a tiny bit of helium vapor inside. The electrons will follow circular orbits inside this tube and collide with and ionize the helium atoms, causing the gas to glow and making the beam visible. 3. Locate the grid and anode inside the glass tube. It is the pair of vertically oriented metal cylinders with a gap between them. At its top and center is the filament or cathode that will be heated by a current to emit the electrons. You should be able to easily see the filament, because it will be brightly lit. 4. There are three separate electrical circuits: (a) Filament/cathode heater (over which you have no control); (b) Accelerating Voltage between cathode and anode;

6 L5-6 Lab 5 Electron Charge-to-Mass Ratio (c) Coil Current for the Helmholtz coils. Figure 5.6: Cathode, Grid and Anode. The electrons are accelerated down from the cathode to the anode. Their initial direction is down and their path of motion will be bent to the left in the above diagram by the magnetic field 5. Measure the diameter of the Helmholtz coils in several places and take the average. Record the mean radius R and the standard error below. Radius R : ± cm 6. Measure the mean separation d between the coils. You may want to average several measurements here also. Coil separation d: cm Verify that d R. (Note: We have found they may not quite agree.) 7. The manufacturer states that there are 130 turns in each coil. 8. Calculate the constant of proportionality between the current passing through the Helmholtz coils and the magnetic field produced. You will need the above parameters to do this. Look at Equation (5.3). Show your work. B Helmholtz (Tesla) = I (amps)

8 L5-8 Lab 5 Electron Charge-to-Mass Ratio 12. Describe what happens to the beam of electrons as the coil current is increased: Question 1-2: What causes this behavior in step 12? 13. Set the coil current to 1.7 A. Adjust the accelerating voltage while looking at the electron beam path. Question 1-3: What do you observe as the accelerating voltage is changed while keeping the coil current constant? Explain why this occurs. 14. While the electron beam is somewhere near the middle of the glass rod, use the bar magnet to see how it affects the electron beam. Question 1-4: Describe what you observe as you move the bar magnet around. Can you produce a helical path for the electron? Can you see why you want to keep spurious magnetic fields away from the electron beam? 15. Make sure you place the bar magnet at the other end of the table from your apparatus to minimize any possible effects.

9 Lab 5 Electron Charge-to-Mass Ratio L5-9 Activity 1-2: Measurement Of Charge-To-Mass Ratio NOTE: The electrons collide with the helium gas atoms that were introduced to make the beam visible. Unavoidably, the electrons lose some energy in the collisions (that make the beam visible). In order to minimize this effect, you should concentrate on those electrons that have the highest energy: those at the outer edge of the beam (largest radius). Ask your TA if you are uncertain about this. 1. We will make measurements for the electrons moving in various circular orbits. You will write down the diameter of the path by reading the illuminated tube. You will need these in order to determine e/m from Equation (5.7). 2. Set the anode voltage to 300 V. Change the coil current until the beam hits the 5 cm mark. This will be the diameter, not the radius. Record the coil current and diameter in Table 5.1. Decrease the coil current in turn to go through each 0.5 cm of the diameter. The 0.5 cm marks are a vertical line between the numbered cm marks. This can proceed quite rapidly by one person observing the beam while changing the coil current and another person filling in Table Proceed in turn to observe and record data for the other accelerating voltages in Table 5.1. Sometimes the beam will physically not be present for every position listed in Table 5.1. Leave the table blank when this occurs. 4. Turn the main power off. Diameter (cm) Accelerating (anode) Voltage 250 V 300 V 350 V 400 V 450 V 500 V Coil Coil Coil Coil Coil Coil Current Current Current Current Current Current (A) (A) (A) (A) (A) (A) Table 5.1: 5. Open up Excel file L05.Table1-1.xls to calculate the values for e/m for each of your data points. Notice that there are two sheets (tabs) for this file (at bottom). You put your Coil

10 L5-10 Lab 5 Electron Charge-to-Mass Ratio Current data into the sheet labeled Data. The sheet labeled e/m uses that data to determine e/m. If you change the format of the data sheet, the e/m calculation will not work properly. Don t forget to insert the constants R and number of turns N. 6. Click on the tab sheet for e m and make sure that the e/m values have been calculated for all the cells with data. Delete the results for those calculations without data. In this case, you will probably see an error message. Average all your experimental results in Excel to obtain a mean value for e/m. Show this result and all subsequent calculations on your Excel file. e/m: x C/kg 7. Find the statistical error of your average value. [Hint: Think standard error of the mean ] Standard error of mean: x C/kg 8. What number is your apparatus? Question 1-5: Discuss how well your result compares with the accepted value of e/m = 1.759x10 11 C/kg within the statistical error. 9. Print out one Excel data table for your group. Do not delete your Excel file. You will need it later. Activity 1-3: Investigation of Systematic Error It is useful in an experiment like this to see if there are systematic errors that might affect your final results. From Equations (5.7) and (5.3) [and a review of Appendix C, of course!] we can see that the relative error in our experimental determination of e/m due to errors in our measured quantities (V, I, R, and r) is given by: ( σe/m e/m ) 2 = ( σv ) 2 ( + 2 σ I V I ) 2 + ( 2 σ R R ) 2 + ( 2 σ r r ) 2 (5.8)

11 Lab 5 Electron Charge-to-Mass Ratio L5-11 Question 1-6: With Equation (5.7) in mind, discuss possible sources of error in your experiment. 1. It is possible that we may learn something about our errors if we compare the values of e/m versus our parameters. Because we have our data in Excel, it is quite easy to do that. 2. Make three plots. Plot all your values of e/m versus each of the following. (a) accelerating voltage, (b) coil current, and (c) orbit radius. Excel Plots: (a) Highlight column of data you want in x-axis (b) Press and hold down Control button while highlighting data you want on the y-axis. (c) Go to Chart Wizzard icon and choose scatter plot (d) Put on labels, etc Place labels and units on the two axes and label the groupings of values on the plot. 3. Print out one copy each of these three graphs and include them in your report. Question 1-7: Look carefully at the plot versus accelerating voltage. Do you see any patterns? Could these possibly indicate any systematic problems? Discuss these possible errors and how you might be able to correct for them or improve the experiment. Question 1-8: Look carefully at the plot versus coil current. Do you see any patterns? Could these possibly indicate any systematic problems? Discuss these possible errors and how you might be able to correct for them or improve the experiment.

12 L5-12 Lab 5 Electron Charge-to-Mass Ratio Question 1-9: Look carefully at the plot versus orbit radius. Do you see any patterns? Could these possibly indicate any systematic problems? Discuss these possible errors and how you might be able to correct for them or improve the experiment. Question 1-10: List any other systematic errors you can think of and discuss. Please clean up your lab area and make sure that the power on the apparatus is off

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